Factoring Expressions with Fractional or Negative Exponents

Expressions with fractional or negative exponents can be factored by pulling out a GCF. Look for the variable or exponent that is common to each term of the expression and pull out that variable or exponent raised to the lowest power. These expressions follow the same factoring rules as those with integer exponents. For instance, $2{x}^{\frac{1}{4}}+5{x}^{\frac{3}{4}}$ can be factored by pulling out ${x}^{\frac{1}{4}}$ and being rewritten as ${x}^{\frac{1}{4}}\left(2+5{x}^{\frac{1}{2}}\right)$ .

Example 7: Factoring an Expression with Fractional or Negative Exponents

Factor $3x{\left(x+2\right)}^{\frac{-1}{3}}+4{\left(x+2\right)}^{\frac{2}{3}}$ .

Solution

Factor out the term with the lowest value of the exponent. In this case, that would be ${\left(x+2\right)}^{-\frac{1}{3}}$ .

\begin{array}{cc} {(x + 2)}^{- \frac{1}{3}} (3 x + 4 (x + 2)) & Factor out the GCF . \\ {(x + 2)}^{- \frac{1}{3}} (3 x + 4 x + 8) & Simplify . \\ {(x + 2)}^{- \frac{1}{3}} (7 x + 8) \end{array}

$\begin{array}{cc}{\left(x+2\right)}^{-\frac{1}{3}}\left(3x+4\left(x+2\right)\right)\hfill & \text{Factor out the GCF}.\hfill \\ {\left(x+2\right)}^{-\frac{1}{3}}\left(3x+4x+8\right)\hfill & \text{Simplify}.\hfill \\ {\left(x+2\right)}^{-\frac{1}{3}}\left(7x+8\right)\hfill & \end{array}$

Try It 8

Factor $2{\left(5a - 1\right)}^{\frac{3}{4}}+7a{\left(5a - 1\right)}^{-\frac{1}{4}}$ .

Solution

Factoring Polynomials